On coding without restrictions for the AWGN channel

Many coded modulation constructions, such as lattice codes, are visualized as restricted subsets of an infinite constellation (IC) of points in the n-dimensional Euclidean space. The author regards an IC as a code without restrictions employed for the AWGN channel. For an IC the concept of coding rate is meaningless and the author uses, instead of coding rate, the normalized logarithmic density (NLD). The maximum value C_∞ such that, for any NLD less than C_∞, it is possible to construct an PC with arbitrarily small decoding error probability, is called the generalized capacity of the AWGN channel without restrictions. The author derives exponential upper and lower bounds for the decoding error probability of an IC, expressed in terms of the NLD. The upper bound is obtained by means of a random coding method and it is very similar to the usual random coding bound for the AWGN channel. The exponents of these upper and lower bounds coincide for high values of the NLD, thereby enabling derivation of the generalized capacity of the AWGN channel without restrictions. It is also shown that the exponent of the random coding bound can be attained by linear ICs (lattices), implying that lattices play the same role with respect to the AWGN channel as linear-codes do with respect to a discrete symmetric channel